| L(s) = 1 | + 3.27e4·2-s + 3.71e6·3-s + 1.07e9·4-s − 3.05e10·5-s + 1.21e11·6-s + 1.17e13·7-s + 3.51e13·8-s − 6.03e14·9-s − 1.00e15·10-s − 9.08e15·11-s + 3.99e15·12-s + 5.73e16·13-s + 3.84e17·14-s − 1.13e17·15-s + 1.15e18·16-s − 8.37e18·17-s − 1.97e19·18-s + 1.80e17·19-s − 3.27e19·20-s + 4.35e19·21-s − 2.97e20·22-s + 2.07e18·23-s + 1.30e20·24-s + 9.31e20·25-s + 1.88e21·26-s − 4.53e21·27-s + 1.25e22·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.149·3-s + 0.5·4-s − 0.447·5-s + 0.105·6-s + 0.933·7-s + 0.353·8-s − 0.977·9-s − 0.316·10-s − 0.655·11-s + 0.0747·12-s + 0.310·13-s + 0.659·14-s − 0.0668·15-s + 0.250·16-s − 0.709·17-s − 0.691·18-s + 0.00273·19-s − 0.223·20-s + 0.139·21-s − 0.463·22-s + 0.00162·23-s + 0.0528·24-s + 0.200·25-s + 0.219·26-s − 0.295·27-s + 0.466·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(32-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s+31/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(16)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{33}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 3.27e4T \) |
| 5 | \( 1 + 3.05e10T \) |
| good | 3 | \( 1 - 3.71e6T + 6.17e14T^{2} \) |
| 7 | \( 1 - 1.17e13T + 1.57e26T^{2} \) |
| 11 | \( 1 + 9.08e15T + 1.91e32T^{2} \) |
| 13 | \( 1 - 5.73e16T + 3.40e34T^{2} \) |
| 17 | \( 1 + 8.37e18T + 1.39e38T^{2} \) |
| 19 | \( 1 - 1.80e17T + 4.37e39T^{2} \) |
| 23 | \( 1 - 2.07e18T + 1.63e42T^{2} \) |
| 29 | \( 1 + 4.65e22T + 2.15e45T^{2} \) |
| 31 | \( 1 + 1.09e23T + 1.70e46T^{2} \) |
| 37 | \( 1 - 1.04e24T + 4.11e48T^{2} \) |
| 41 | \( 1 + 1.45e25T + 9.91e49T^{2} \) |
| 43 | \( 1 + 6.31e24T + 4.34e50T^{2} \) |
| 47 | \( 1 - 7.71e25T + 6.83e51T^{2} \) |
| 53 | \( 1 - 2.22e25T + 2.83e53T^{2} \) |
| 59 | \( 1 + 4.61e27T + 7.87e54T^{2} \) |
| 61 | \( 1 + 1.68e27T + 2.21e55T^{2} \) |
| 67 | \( 1 + 2.09e28T + 4.05e56T^{2} \) |
| 71 | \( 1 + 7.31e28T + 2.44e57T^{2} \) |
| 73 | \( 1 + 1.79e28T + 5.79e57T^{2} \) |
| 79 | \( 1 + 3.55e29T + 6.70e58T^{2} \) |
| 83 | \( 1 + 1.38e29T + 3.10e59T^{2} \) |
| 89 | \( 1 + 2.03e30T + 2.69e60T^{2} \) |
| 97 | \( 1 - 8.65e30T + 3.88e61T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.25129642095814465644907896598, −11.70759852142898920607028920529, −10.83952189156692821548815250398, −8.680083224204161246746740425056, −7.50740521851980776096344695676, −5.77417643163608552608697863789, −4.58289123684851656691846756273, −3.16517723602498177324474627572, −1.83435827452413777695292052411, 0,
1.83435827452413777695292052411, 3.16517723602498177324474627572, 4.58289123684851656691846756273, 5.77417643163608552608697863789, 7.50740521851980776096344695676, 8.680083224204161246746740425056, 10.83952189156692821548815250398, 11.70759852142898920607028920529, 13.25129642095814465644907896598
